Summary: Automata for arithmetic Meyer sets ?
Shigeki Akiyama 1 , Frederique Bassino 2 , and Christiane Frougny 3
1 Department of Mathematics, Faculty of Sciences, Niigata University, Ikarashi-2,
8050, Niigata 950-2181, Japan, akiyama@math.sc.niigata-u.ac.jp
2 Institut Gaspard Monge, Universite de Marne-la-Vallee, 5 Boulevard Descartes,
Champs-sur-Marne 77454 Marne-la-Vallee Cedex 2, France, bassino@univ-mlv.fr
3 LIAFA, UMR 7089, 2 Place Jussieu, 75251 Paris Cedex 05, France, and
Universite Paris 8, Christiane.Frougny@liafa.jussieu.fr
Abstract. The set Z of -integers is a Meyer set when is a Pisot
number, and thus there exists a nite set F such that Z Z Z+F .
We give nite automata describing the expansions of the elements of
Z and of Z Z . We present a construction of a such a nite set F ,
and a method to minimize the size of F . We obtain in this way a nite
transducer that performs the decomposition of the elements of Z Z
as a sum belonging to Z + F .
1 Introduction
The so-called Meyer sets have been introduced by Meyer [11, 12] under the name
of \quasicrystals" in order to formalize the quasicrystals discovered by the physi-
cists in the eighties. A set X is a Delaunay set if it is uniformly discrete and
relatively dense. A set X is a Meyer set if it is a Delaunay set and there exists